3.3 Product Measures and Fubini's Theorem
Having established integration theory in one dimension (or on a single abstract space), we naturally need to extend integration to products of multiple dimensions or multiple spaces (e.g., transitioning from single-variable integrals to multiple integrals). This requires introducing the concept of product measures and addressing a central question: Can a multiple integral be reduced to iterated integrals? And Can the order of integration be exchanged?
These are precisely the questions answered by the famous Tonelli theorem and Fubini theorem. Throughout this discussion, we assume that the involved measure spaces are \(\sigma\)-finite.
1. Product Measurable Spaces and Sections
To define a measure on the product space \(X \times Y\), we first need to establish a \(\sigma\)-algebra on it. The basic building blocks are "measurable rectangles".
Definition 3.3.1 (Measurable Rectangle and Product \(\sigma\)-Algebra)
Let \((X, \mathcal{M})\) and \((Y, \mathcal{N})\) be two measurable spaces. A set \(A \times B\) is called a measurable rectangle if \(A \in \mathcal{M}\) and \(B \in \mathcal{N}\).
The family of all measurable rectangles generates the product \(\sigma\)-algebra on \(X \times Y\), denoted by:
When dealing with functions and sets of two variables, the "sections" obtained by fixing one variable are important objects of study.
Definition 3.3.2 (Sections of Sets and Functions)
For \(E \subset X \times Y\), fix \(x \in X\) or \(y \in Y\), and define its set sections as:
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\(x\)-section: \(E_x = \{y \in Y \mid (x, y) \in E\} \subset Y\)
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\(y\)-section: \(E^y = \{x \in X \mid (x, y) \in E\} \subset X\)
For a function \(f\) defined on \(X \times Y\), define its function sections as:
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\(f_x(y) = f(x, y)\) (viewed as a function defined on \(Y\))
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\(f^y(x) = f(x, y)\) (viewed as a function defined on \(X\))
Theorem 3.3.1 (Measurability of Sections)
If \(E \in \mathcal{M} \otimes \mathcal{N}\), then for all \(x \in X\) and \(y \in Y\), we have \(E_x \in \mathcal{N}\) and \(E^y \in \mathcal{M}\).
If \(f\) is an \(\mathcal{M} \otimes \mathcal{N}\)-measurable function on \(X \times Y\), then \(f_x\) is \(\mathcal{N}\)-measurable and \(f^y\) is \(\mathcal{M}\)-measurable.
Proof of Measurability of Set Sections
We prove that \(E_x \in \mathcal{N}\). Construct the family:
First, if \(E = A \times B\) is a measurable rectangle, then when \(x \in A\), \(E_x = B \in \mathcal{N}\); when \(x \notin A\), \(E_x = \emptyset \in \mathcal{N}\). Hence all measurable rectangles belong to \(\mathcal{R}\).
Next, we verify that \(\mathcal{R}\) is a \(\sigma\)-algebra.
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Closed under complements: \((E^c)_x = (E_x)^c\). Since \(\mathcal{N}\) is a \(\sigma\)-algebra, if \(E_x \in \mathcal{N}\), then \((E_x)^c \in \mathcal{N}\), so \(E^c \in \mathcal{R}\).
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Closed under countable unions: \((\bigcup_{j=1}^\infty E_j)_x = \bigcup_{j=1}^\infty (E_j)_x\). By the closure properties of \(\mathcal{N}\), the union is also in \(\mathcal{R}\).
Since \(\mathcal{R}\) is a \(\sigma\)-algebra containing all measurable rectangles, and \(\mathcal{M} \otimes \mathcal{N}\) is the smallest \(\sigma\)-algebra containing measurable rectangles, we must have \(\mathcal{M} \otimes \mathcal{N} \subset \mathcal{R}\). The theorem is proved.
2. Monotone Class Theorem
When proving properties of product measures, because \(\sigma\)-algebras are quite complicated, we often first prove the properties on an algebra and then extend them to the whole \(\sigma\)-algebra via the "monotone class theorem". This is an extremely powerful proof tool.
Definition 3.3.3 (Monotone Class)
Let \(\mathcal{C} \subset 2^X\) be a family of sets. \(\mathcal{C}\) is called a monotone class if it satisfies:
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For an increasing sequence \(E_1 \subset E_2 \subset \dots\), if \(E_j \in \mathcal{C}\), then \(\bigcup_{j=1}^\infty E_j \in \mathcal{C}\).
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For a decreasing sequence \(E_1 \supset E_2 \supset \dots\), if \(E_j \in \mathcal{C}\), then \(\bigcap_{j=1}^\infty E_j \in \mathcal{C}\).
Obviously, any \(\sigma\)-algebra is a monotone class. Conversely, a family that is both an algebra and a monotone class must be a \(\sigma\)-algebra.
Theorem 3.3.2 (Monotone Class Theorem)
Let \(\mathcal{A}\) be an algebra. If \(\mathcal{C}\) is a monotone class containing \(\mathcal{A}\), then \(\mathcal{C}\) must contain the \(\sigma\)-algebra generated by \(\mathcal{A}\), i.e.,
3. Construction of Product Measures
Now we can define the product of two measures.
Theorem 3.3.3 (Section Integrals and Product Measure)
Let \((X, \mathcal{M}, \mu)\) and \((Y, \mathcal{N}, \nu)\) be \(\sigma\)-finite measure spaces. For any \(E \in \mathcal{M} \otimes \mathcal{N}\):
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The map \(x \mapsto \nu(E_x)\) is \(\mathcal{M}\)-measurable;
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The map \(y \mapsto \mu(E^y)\) is \(\mathcal{N}\)-measurable;
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And their integrals are equal, jointly defining the product measure \(\mu \times \nu\) on \(X \times Y\):
Sketch of Proof of Theorem 3.3.3
The core of the proof uses the Monotone Class Theorem.
First, if \(E = A \times B\) is a measurable rectangle, then \(\nu(E_x) = \nu(B) \cdot \chi_A(x)\). This is a simple function, clearly measurable, and its integral is \(\mu(A)\nu(B)\). Symmetrically, \(\int \mu(E^y)d\nu(y) = \mu(A)\nu(B)\) also holds.
Collect all sets \(E \in \mathcal{M} \otimes \mathcal{N}\) that satisfy the above properties, forming the family \(\mathcal{C}\). By the discussion above, all finite disjoint unions of rectangles (i.e., the algebra \(\mathcal{A}\) generated by rectangles) are in \(\mathcal{C}\).
Since the measure spaces are \(\sigma\)-finite, we can use continuity from below (for increasing sequences) and continuity from above (using the Dominated Convergence Theorem for decreasing sequences) to show that \(\mathcal{C}\) has the monotone class property.
By the Monotone Class Theorem, the monotone class \(\mathcal{C}\) containing the algebra \(\mathcal{A}\) must contain \(\sigma(\mathcal{A}) = \mathcal{M} \otimes \mathcal{N}\). Hence the property holds for all measurable sets.
4. Tonelli's Theorem and Fubini's Theorem
These are the two core theorems for computing multiple integrals. Tonelli's theorem deals with nonโnegative functions, while Fubini's theorem deals with general integrable functions. They are often used together.
Theorem 3.3.4 (Tonelli's Theorem: For Nonโnegative Functions)
Let \((X, \mathcal{M}, \mu)\) and \((Y, \mathcal{N}, \nu)\) be \(\sigma\)-finite measure spaces. If \(f \in L^+(X \times Y)\) (i.e., \(f\) is a nonโnegative measurable function), then:
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The functions \(g(x) = \int_Y f_x \, d\nu\) and \(h(y) = \int_X f^y \, d\mu\) are nonโnegative measurable functions on \(X\) and \(Y\), respectively.
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The double integral of \(f\) can be computed via iterated integrals in any order (even if the result is \(+\infty\)):
Derivation of Tonelli's Theorem
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Characteristic function: If \(f = \chi_E\), then by the definition of product measure (Theorem 3.3.3), the theorem holds directly.
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Nonโnegative simple function: If \(f = \sum a_j \chi_{E_j}\), by linearity of the integral, the theorem holds.
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Nonโnegative measurable function: By the Simple Function Approximation Theorem, there exists a monotonically increasing sequence of nonโnegative simple functions \(\phi_n \uparrow f\). For each \(\phi_n\), the iterated integral equals the double integral. Applying the Monotone Convergence Theorem (MCT) to both sides, exchanging limit and integral, we obtain that the theorem holds for any nonโnegative function \(f\).
Theorem 3.3.5 (Fubini's Theorem: For Absolutely Integrable Functions)
Let \((X, \mathcal{M}, \mu)\) and \((Y, \mathcal{N}, \nu)\) be \(\sigma\)-finite measure spaces. If \(f \in L^1(\mu \times \nu)\) (i.e., \(f\) is absolutely integrable on the product space), then:
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For almost every \(x \in X\), \(f_x \in L^1(\nu)\) (i.e., the section function is integrable); for almost every \(y \in Y\), \(f^y \in L^1(\mu)\).
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The functions \(g(x) = \int_Y f_x \, d\nu\) and \(h(y) = \int_X f^y \, d\mu\), obtained by integrating the sections, are integrable on \(X\) and \(Y\) respectively.
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The double integral equals the iterated integrals:
Proof Idea for Fubini's Theorem
Since \(f \in L^1\), we have \(\int |f| \, d(\mu \times \nu) < \infty\).
Because \(|f|\) is a nonโnegative measurable function, we can apply Tonelli's theorem to it:
This shows that the outer integral is finite. By the "almost everywhere" property of integrals of nonโnegative functions, the integrand \(\int_Y |f_x| d\nu\) must be finite almost everywhere on \(X\). This proves \(f_x \in L^1(\nu)\) a.e.
Finally, decompose the real function \(f\) into its positive and negative parts \(f = f^+ - f^-\) (for complex functions, decompose into real/imaginary parts and then positive/negative parts). Since \(|f|\) is integrable, both \(f^+\) and \(f^-\) are integrable on \(X \times Y\). Apply Tonelli's theorem to \(f^+\) and \(f^-\) separately, then subtract using linearity of the integral, proving Fubini's theorem.
Summary of usage strategy: In practical computations, we often first apply Tonelli's theorem to \(|f|\) to verify whether the integral is finite. If \(\int \int |f| d\mu d\nu < \infty\), then \(f \in L^1(\mu \times \nu)\), and we can safely use Fubini's theorem to exchange the order of integration for \(f\) itself.
5. A Brief Note on Lebesgue Integration in \(\mathbb{R}^n\)
Based on the theory of product measures, integration theory in highโdimensional Euclidean space \(\mathbb{R}^n\) becomes very natural.
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Product of Lebesgue measures: The Borel measure on \(\mathbb{R}^n\) can be viewed as the product of \(n\) oneโdimensional Borel measures. After completing it, we obtain the Lebesgue measure \(m_n\) on \(\mathbb{R}^n\).
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Outer measure approximation: For any measurable set \(E \subset \mathbb{R}^n\) and its Lebesgue measure \(m_n(E)\), we can approximate it from the outside by open sets (\(\inf \{m(U) \mid U \supset E\}\)) and from the inside by compact sets (\(\sup \{m(K) \mid K \subset E\}\)).
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Geometric invariance: In the oneโdimensional case, we know Lebesgue measure is translation invariant. In \(\mathbb{R}^n\), the Lebesgue measure \(m_n\) is invariant not only under translations but also under rotations in space. This gives Lebesgue integration a natural advantage when dealing with geometric and physical problems.